1888 - 89 .] Dr Sang on Airs Resistance to Oscillating Bodies. 183 
we may, then, regard the ratio of 2000 to 1 as not too high. Thus, 
in all cases of any importance to us, we may regard Q as a very 
small fraction whose higher powers need not be taken into account ; 
it may be more than enough for us to retain the second power. 
We have now to examine the signs of P and Q ; let us consider 
first that of P. 
When the oscillating body is moving from A on the side to which 
we have agreed to attach the sign + x , towards the middle point 0, 
+ x is decreasing and its derivative v must take the sign - , at the 
same time that velocity is increasing or becoming more - , where- 
fore lt v is - , and so therefore must be P. When the body has 
passed O, x is increasing in the - direction, v therefore is still — ; it 
is, however, now decreasing, its derivative is therefore + , so that Vx 
must have the sign + , but then x is — , wherefore P must be - in 
this quarter of the oscillation. When the body has reached the 
extreme limit B and is returning along BO, its - x is decreasing more 
and more rapidly; that is to say, both its velocity and the increment 
of its velocity take the sign + , Vx then is + , but x is - , wherefore 
P must be - in this third quarter also. Lastly, when the body has 
passed 0 into the quarter OA, its + velocity is decreasing, and both 
~Px and P take the sign - . Thus it seems that, in all the four 
quarters of the oscillation, P must have the sign — ; we shall then 
write it as - a 2 . 
The air’s resistance, denoted by Q u 2 , tends to lessen the velocity, 
to hinder acceleration, to help retardation irrespective of signs. 
When the body is moving from A towards O, the - velocity is being 
increased in the — direction by - a?x, and this increase is opposed by 
Qv 2 , wherefore Q here takes the sign +, and the total influence 
is - a 2 x + Qv 2 . In the second quarter, that is from 0 to B, the 
- velocity is being lessened, and the air’s resistance helps this, where- 
fore Qv 2 takes here also the sign + . Again, in the third quarter, 
from B towards 0, the + velocity is being augmented, and the 
resistance opposing this augmentation must take the sign - . Also, 
in the fourth quarter, the + velocity is being retarded, the retarda- 
